1. Technical Field
The present invention is broadly directed to the characterization of transducers and the use of a characterized transducer in the investigation of materials. More particularly, the present invention is directed to using a characterized transducer in the measurement of the impudence of cement located behind a section of casing in a borehole.
Characterization of transducers is important in applications of ultrasonic technology. Manufacturers of transducers typically specify various characteristics of their transducers, including the center frequency and bandwith of the transducers which describes the time response of the transducer, and the f number, focal length, or transducer radius which describe the acoustic field of the transducer in space. As used in the art, the term "radius" refers to the radius of the electrode of the piezoelectric transducer which frequently is also the radius of the piezoelectric ceramic portion of the transducer.
Another spatial field characteristic of transducers is known in the art as "standoff". By exciting a transducer with an impulse response and analyzing the return signal as reflected off a reflector, the transducer's frequency characteristics are determined. Standoff refers to the spacing between the transducer and the reflector when the transducer is tested in this pulse-echo manner.
As used herein, "effective radius" refers to the dispersion of acoustic energy from the transducer being characterized; and more particularly to the radius of a well-defined transducer which would produce nearly the same spatial acoustic field as that of the transducer being characterized. A relatively small effective radius corresponds to an acoustic field that has energy distributed over a wide range of wavenumbers. Conversely, a relatively large effective radius corresponds to a transducer having energy distributed over a relatively narrower range of wavenumber.
Additionally, as used herein, "effective standoff" refers to the shape of the wavefronts generated by the transducer upon excitation; and more particularly to the standoff of a well-defined transducer which would produce nearly the same spatial field as that for the transducer being characterized. The effective standoff indicates the phasing of the transducer acoustic field at different wavenumbers. A relatively small effective standoff corresponds to an acoustic field having a relatively small radius of curvature in space of equal phase points. Conversely, a relatively large effective standoff corresponds to an acoustic field in which the equal phase points have a relatively large radius of curvature.
2. The State of the Art
In a typical wellbore application, cement is placed between the subsurface formation exposed by a borehole and a casing placed therein. The cement hydraulically isolates the different zones, e.g. water, oil, and gas, which may be located in the subsurface formation. Without a cement seal, the fluids under pressure in one zone may flow between the borehole and the casing to other zones. Thus, production zones could become contaminated, or the fluid of interest could escape from the zone determined for production, thereby rendering production uneconomical. Accordingly, it is imperative that there be a good cement seal; i.e. high cement quality between the formation and casing.
Defects in cement quality include annuli, channels, as well as complete voids in the cement. Defects in cement quality can also occur due to improper hardening of the cement, e.g., due to an improper amount of water in the cement mixture. Defects in cement quality can also occur during the life of the well such as due to corrosion of the casing. Once casing is installed in a well, it is at best difficult to remove for above-ground inspection. Thus it is desirable to be able to check the cement quality of the casing in situ.
Techniques for determining the cement quality of situ are well known. For example, U.S. Pat. No. 4,255,798 to Havira, and assigned to the same assignee as the present invention, which patent is hereby incorporated by reference herein, employs an acoustic pulse-echo technique having either a single transducer capable of directing its pulse at various azimuths or a plurality of transducers azimuthally located about a tool. The transducer transmits a pulse towards the casing and receives the return signal therefrom. The received signal includes an initial reflection segment which results largely from the reflection of the pulse off the inner surface of the casing, and a reverberation segment which results largely from the subsequent reverberations of the resonating casing section. The reverberation segment is indicative of the energy of the echo produced by the casing-cement interface.
In the preferred embodiment of the Havira patent, the initial reflection segment and the reverberation segment are processed separately. The entire initial reflection segment is selected to determined the peak value thereof. Thereafter, a predetermined portion of the reverberation segment is selected, allowing the energy of the reverberation segment to be calculated by an integrator. The calculated reverberation segment energy is divided by the peak value to generate a normalized cement quality signal.
It is has been found that the wideband signal processing with fixed time windows which is utilized in the Havira patent leads to several inaccuracies in the determination of cement quality. For example, inaccuracies are introduced due to the inclusion of unwanted noise components and phase variation between resonances, as well as due to the fact that the temporal portion of the reverberation segment signal indicative of cement quality varies with casing thickness. Additionally, it has been found that the information indicative of cement quality is located in a narrow frequency band about the frequency of the casing's thickness resonance in the reverberation segment. The narrow frequency band varies according to casing thickness.
U.S. Pat. No. 4,928,269 issued to Kimball, et al., assigned to the same assignee hereof, and hereby incorporated by reference herein, teaches a technique for determining the cement quality in situ wherein the problems inherent with wideband signal processing and fixed time windows are obviated. Specifically, the return waveform is analyzed to determine the casing thickness resonance frequency, and the return waveform is bandpass filtered about the determined thickness resonance frequency to the substantial exclusion of other thickness resonances. After time windowing a portion of the reverberation segment based on the casing thickness resonant frequency, the energy content therein is calculated.
In the Kimball et al. patent, a portion of the initial reflection segment is also time windowed, and the energy content therein is calculated. This energy calculation, representative of the energy of the acoustic reflection from the inner surface of the casing section is used to normalize the cementation signal calculated from the reverberation segment. Further, a calibration signal, commonly referred to as the free-pipe value, is determined by firing a transducer in pulse-echo fashion towards a casing, except that the impedance of the material behind the section of casing is known. After taking the difference between the normalized cementation signal and the normalized calibration signal, the impedance of the cement behind the casing is determined. By comparing the value of the impedance to predetermined impedance values indicative of good and bad cement quality, the cement quality can be determined.
It is known in the art that the particulars of the waveform returning from the casing as a result of an acoustic transducer pulse are a function of several factors, including transducer parameters. Various models which approximate the waveform as a function of the factors and parameters have been proposed. One of those models assumes that the transducer acoustic field has a single plane wave oriented at normal incidence to a planar reflecting surface. It is well known, however, that the normal incidence planar model doe not accurately describe the response as seen by the transducer. See e.g. Johnson, R. K. and Devaney, A. J. "Transducer Effects in Acoustic Scattering Measurements," 41 Applied Physics Letters, 622-24 (October, 1982), which is hereby incorporated by reference herein. These inaccuracies are evident in differences between waveforms from the normal incidence planar model and those obtained experimentally. Without calibration, the techniques disclosed in U.S. Pat. Nos. 4,255,798 and 4,928,269 which are based on the normal incidence planar model can be off by as much as 1 MegaRayl (MRayl). A 1 MRayl cement impedance error can be important, particularly when modern light-weight cements are in use.
A second model which better approximates the return waveform is set forth the in the aforementioned article by Johnson and Devaney. The "Johnson-Devaney" model equates the return waveform V(f) according to: ##EQU1## where W(f,.theta.) is a kernel, and R(f,.theta..vertline.z) are reflection coefficients of the target. The Johnson-Devaney model assumes that the transducer is circularly symmetric and that the target is a flat surface. Thus, even if Johnson and Devaney had provided a method for utilizing their model to obtain measurements of cement impedance (which they did not), the model would have inaccuracies due to the fact that the reflective surface downhole is a casing which is curved.
A third model for approximating a return waveform is set forth in C. J. Randall, and F. E. Stanke, "Mathematical Model for Ultransonic, Internal Inspection of Cylindrically Layered Structures", J. Acoustical Soc. Am. 83; pp. 1295-1305 (1988), which develops a complex equation for the waveform received by an irregular transducer as a result of a reflection from a curved surface. In the Randall-Stanke model, the return wave is a function of the casing diameter, transducer parameters such as length and apodization, and stand-off between the transducer and the target. While the Randall-Stanke model is theoretically a very good approximation, the calculation time per waveform is very large. Also, the large number of unknown parameters makes an inversion difficult, as a search over many possible alternative parameter values would be required. Thus, the Randall-Stanke model is not particularly useful in developing a solution to the forward problem of finding cement impedance.